Geodesic
New extension of the variational McShane integral of vector-valued functions
Mathematica Bohemica, Tome 144 (2019) no. 2, pp. 137-148
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We define the Hake-variational McShane integral of Banach space valued functions defined on an open and bounded subset $G$ of $m$-dimensional Euclidean space $\mathbb {R}^{m}$. It is a "natural" extension of the variational McShane integral (the strong McShane integral) from $m$-dimensional closed non-degenerate intervals to open and bounded subsets of $\mathbb {R}^{m}$. We will show a theorem that characterizes the Hake-variational McShane integral in terms of the variational McShane integral. This theorem reduces the study of our integral to the study of the variational McShane integral. As an application, a full descriptive characterization of the Hake-variational McShane integral is presented in terms of the cubic derivative.
We define the Hake-variational McShane integral of Banach space valued functions defined on an open and bounded subset $G$ of $m$-dimensional Euclidean space $\mathbb {R}^{m}$. It is a "natural" extension of the variational McShane integral (the strong McShane integral) from $m$-dimensional closed non-degenerate intervals to open and bounded subsets of $\mathbb {R}^{m}$. We will show a theorem that characterizes the Hake-variational McShane integral in terms of the variational McShane integral. This theorem reduces the study of our integral to the study of the variational McShane integral. As an application, a full descriptive characterization of the Hake-variational McShane integral is presented in terms of the cubic derivative.
DOI : 10.21136/MB.2018.0114-17
Classification : 28B05, 46B25, 46G10
Keywords: Hake-variational McShane integral; variational McShane integral; Banach space; $m$-dimensional Euclidean space 
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Kaliaj, Sokol Bush. New extension of the variational McShane integral of vector-valued functions. Mathematica Bohemica, Tome 144 (2019) no. 2, pp. 137-148. doi: 10.21136/MB.2018.0114-17

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